Pre-Calculus: 1,001 Practice Problems For Dummies (+ Free Online Practice) (For Dummies

2014·9.92 MB·English

by  Mary Jane Sterling

1,001 Pre- Calculus
Practice Problems
by Mary Jane Sterling
1,001 Pre-Calculus Practice Problems For Dummies®
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2014 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Control Number: 2014936392
ISBN 978-1-118-85332-0 (pbk); ISBN 978-1-118-85281-1 (ebk); ISBN 978-1-118-85334-4 (ebk)
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents at a Glance
Introduction ............................................................................1
Part I: The Questions ................................................................7
Chapter 1: Getting Started with Algebra Basics ................................................................................9
Chapter 2: Solving Some Equations and Inequalities .....................................................................15
Chapter 3: Function Basics ................................................................................................................21
Chapter 4: Graphing and Transforming Functions .........................................................................29
Chapter 5: Polynomials ......................................................................................................................37
Chapter 6: Exponential and Logarithmic Functions .......................................................................45
Chapter 7: Trigonometry Basics .......................................................................................................53
Chapter 8: Graphing Trig Functions .................................................................................................61
Chapter 9: Get ting Star ted with Trig Identities ..............................................................................67
Chapter 10: Continuing with Trig Identities ....................................................................................73
Chapter 11: Working with Triangles and Trigonometry ................................................................79
Chapter 12: Complex Numbers and Polar Coordinates .................................................................89
Chapter 13: Conic Sections ................................................................................................................97
Chapter 14: Systems of Equations and Inequalities .....................................................................103
Chapter 15: Sequences and Series ..................................................................................................111
Chapter 16: Introducing Limits and Continuity ............................................................................117
Part II: The Answers ............................................................125
Chapter 17: Answers.........................................................................................................................127
Index ..................................................................................527

Table of Contents
Introduction ................................................................. 1
What You’ll Find ..............................................................................................1
How This Workbook Is Organized .................................................................2
Part I: The Questions .............................................................................2
Part II: The Answers ..............................................................................3
Beyond the Book .............................................................................................3
What you’ll find online ..........................................................................3
How to register .......................................................................................4
Where to Go for Additional Help ...................................................................4
Part I: The Questions .................................................... 7
Chapter 1: Getting Started with Algebra Basics . . . . . . . . . . . . . . . . . . .9
The Problems You’ll Work On .......................................................................9
What to Watch Out For ...................................................................................9
Identifying Which System or Systems a Number Belongs To ..................10
Recognizing Properties of Number Systems ..............................................10
Simplifying Expressions with the Order of Operations ............................11
Graphing Inequalities ....................................................................................12
Using Graphing Formulas .............................................................................13
Applying Graphing Formulas .......................................................................13
Chapter 2: Solving Some Equations and Inequalities . . . . . . . . . . . . . .15
The Problems You’ll Work On .....................................................................15
What to Watch Out For .................................................................................15
Using Interval and Inequality Notation .......................................................16
Solving Linear Inequalities ...........................................................................17
Solving Quadratic Inequalities .....................................................................17
Solving Absolute Value Inequalities ............................................................17
Working with Radicals and Fractional Notation ........................................18
Performing Operations Using Fractional Exponents ................................18
Factoring Using Fractional Notation ...........................................................19
Solving Radical Equations ............................................................................19
Rationalizing Denominators .........................................................................20
Chapter 3: Function Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
The Problems You’ll Work On .....................................................................21
What to Watch Out For .................................................................................21
Using Function Notation to Evaluate Function Values .............................22
Determining the Domain and Range of a Function ....................................22
Recognizing Even Functions ........................................................................23
Identifying Odd Functions ............................................................................23
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1,001 Pre-Calculus Practice Problems For Dummies
Ruling Out Even and Odd Functions ...........................................................23
Recognizing One-to-One Functions from Given Relations ........................23
Identifying One-to-One Functions from Equations ....................................25
Recognizing a Function’s Inverse ................................................................25
Determining a Function’s Inverse ................................................................26
Executing Operations on Functions ............................................................26
Performing Function Composition ..............................................................27
Doing More Function Composition .............................................................27
Using the Difference Quotient ......................................................................28
Chapter 4: Graphing and Transforming Functions . . . . . . . . . . . . . . . . .29
The Problems You’ll Work On .....................................................................29
What to Watch Out For .................................................................................29
Functions and Their Inverses ......................................................................30
Sketching Quadratic Functions from Their Equations .............................30
Writing Equations from Graphs of Parabolas ............................................31
Investigating and Graphing Radical Functions ..........................................32
Investigating Absolute Value Functions .....................................................33
Investigating the Graphs of Polynomial Functions ...................................33
Investigating Rational Functions .................................................................34
Transformation of Functions .......................................................................34
Transforming Selected Points Using Functions .........................................34
Sketching Graphs Using Basic Functions and Transformations ..............35
Sketching More Graphs Using Basic Functions and Transformations ......35
Chapter 5: Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
The Problems You’ll Work On .....................................................................37
What to Watch Out For .................................................................................37
Using Factoring to Solve Quadratic Equations ..........................................38
Solving Quadratic Equations by Using the Quadratic Formula ...............38
Using Completing the Square to Solve Quadratic Equations ...................39
Solving Polynomial Equations for Intercepts .............................................39
Using Factoring by Grouping to Solve Polynomial Equations .................40
Applying Descartes’s Rule of Signs .............................................................40
Listing Possible Roots of a Polynomial Equation ......................................40
Dividing Polynomials ....................................................................................41
Using Synthetic Division to Divide Polynomials ........................................41
Checking for Roots of a Polynomial by Using Synthetic Division .............41
Writing Polynomial Expressions from Given Roots ..................................42
Writing Polynomial Expressions When Given Roots and a Point ............42
Graphing Polynomials ...................................................................................43
Writing Equations from Graphs of Polynomials ........................................43
Chapter 6: Exponential and Logarithmic Functions . . . . . . . . . . . . . . . .45
The Problems You’ll Work On .....................................................................45
What to Watch Out For .................................................................................45
Understanding Function Notation ...............................................................46
Graphing Exponential Functions .................................................................46
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Table of Contents
Solving Exponential Equations ....................................................................47
Using the Equivalence bx = y ⇔ log y = x to Rewrite Expressions .........48
b
Using the Equivalence log y = x ⇔ bx = y to Rewrite Expressions .........48
b
Rewriting Logarithmic Expressions ............................................................48
Rewriting Logs of Products and Quotients as Sums and Differences .....49
Solving Logarithmic Equations ....................................................................49
Applying Function Transformations to Log Functions .............................50
Applying Logarithms to Everyday Life .......................................................51
Chapter 7: Trigonometry Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
The Problems You’ll Work On .....................................................................53
What to Watch Out For .................................................................................53
Using Right Triangles to Determine Trig Functions ..................................54
Solving Problems by Using Right Triangles and Their Functions ...........55
Working with Special Right Triangles .........................................................56
Changing Radians to Degrees ......................................................................57
Changing Degrees to Radians ......................................................................57
Finding Angle Measures (in Degrees) in Standard Position .....................57
Determining Angle Measures (in Radians) in Standard Position ............58
Identifying Reference Angles ........................................................................58
Determining Trig Functions by Using the Unit Circle ...............................58
Calculating Trig Functions by Using Other Functions
and Terminal Side Positions .....................................................................59
Using the Arc Length Formula .....................................................................59
Evaluating Inverse Functions .......................................................................60
Solving Trig Equations for x in Degrees .....................................................60
Calculating Trig Equations for x in Radians ...............................................60
Chapter 8: Graphing Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
The Problems You’ll Work On .....................................................................61
What to Watch Out For .................................................................................61
Recognizing Basic Trig Graphs ....................................................................62
Graphing Sine and Cosine .............................................................................64
Applying Function Transformations to Graphs of Trig Functions ..........64
Writing New Trig Functions Using Transformations ................................64
Graphing Tangent and Cotangent ...............................................................65
Interpreting Transformations of Trig Functions .......................................65
Graphing Secant and Cosecant ....................................................................66
Interpreting Transformations from Function Rules ..................................66
Chapter 9: Get ting Star ted with Trig Identities . . . . . . . . . . . . . . . . . . .67
The Problems You’ll Work On .....................................................................67
What to Watch Out For .................................................................................67
Proving Basic Trig Identities ........................................................................68
Returning to Basic Sine and Cosine to Solve Identities ............................69
Using Multiplication by a Conjugate to Solve Identities ...........................70
Solving Identities After Raising a Binomial to a Power .............................70
Solving Identities After Factoring out a Common Function .....................70
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1,001 Pre-Calculus Practice Problems For Dummies
Solving Identities After Combining Fractions ............................................71
Performing Algebraic Processes to Make Identities More Solvable .......71
Chapter 10: Continuing with Trig Identities . . . . . . . . . . . . . . . . . . . . . . .73
The Problems You’ll Work On .....................................................................73
What to Watch Out For .................................................................................73
Using Identities That Add or Subtract Angle Measures ...........................74
Confirming Double-Angle Identities ............................................................74
Using Identities That Double the Size of the Angle ...................................74
Confirming the Statements of Multiple-Angle Identities ...........................74
Creating Half-Angle Identities from Double-Angle Identities .....................75
Creating a Half-Angle Identity for Tangent .................................................75
Using Half-Angle Identities to Simplify Expressions ..................................75
Creating Products of Trig Functions from Sums and Differences ...........75
Using Product-to-Sum Identities to Evaluate Expressions .......................75
Using Sum-to-Product Identities to Evaluate Expressions .......................76
Applying Power-Reducing Identities ...........................................................76
Using Identities to Determine Values of Functions at Various Angles ......76
Working through Identities Using Multiple Methods ................................77
Chapter 11: Working with Triangles and Trigonometry . . . . . . . . . . . .79
The Problems You’ll Work On .....................................................................79
What to Watch Out For .................................................................................79
Applying the Law of Sines to Find Sides .....................................................80
Utilizing the Law of Sines to Find Angles ....................................................80
Using the Law of Sines for Practical Applications .....................................81
Investigating the Ambiguous Case of the Law of Sines .............................81
Determining All Angles and Sides of a Triangle .........................................82
Finding Side Measures by Using the Law of Cosines ................................82
Using the Law of Cosines to Determine an Angle ......................................82
Applying the Law of Cosines to Real-World Situations .............................83
Finding Areas of Triangles by Using the Sine .............................................83
Applying the Trig Formula for Area of a Triangle .....................................84
Using the Trig Formula for Area in Various Situations .............................84
Solving Area Problems Needing Additional Computations ......................85
Finding Areas of Triangles by Using Heron’s Formula ..............................86
Applying Heron’s Formula ............................................................................86
Practical Applications Using Heron’s Formula ..........................................87
Tackling Practical Applications by Using Triangular Formulas ..............87
Chapter 12: Complex Numbers and Polar Coordinates . . . . . . . . . . . . .89
The Problems You’ll Work On .....................................................................89
What to Watch Out For .................................................................................89
Writing Powers of i in Their Simplest Form ...............................................90
Adding and Subtracting Complex Numbers ...............................................90
Multiplying Complex Numbers ....................................................................91
Using Multiplication to Divide Complex Numbers ....................................91
Solving Quadratic Equations with Complex Solutions .............................92
Graphing Complex Numbers ........................................................................92
Identifying Points with Polar Coordinates .................................................94